As Daniel says, I think you are mixing up what is $ T $ and what is $ A $, what is the Random Variable, what is the estimator etc.
Let me tell you my understanding of estimating a Random Variable $VS$ estimating a parameters (frequentist point of view). In both cases I am going to consider that my objective is to minimize the $MSE$:
Let be $X$ a random Variable. If I would want to predict X with a constant minimizing the $MSE$:
$$
x_{mse} = arg\ min _{c} \ E[(X-c)^2] => x_{mse} = E[X] => MSE_{min}= Var(X)
$$
Of course when you are working with data, most of the times you don't know the parameters of the R.V $ X $, in this case for example you don't know the mean so you would have to estimate it. This is to estimate a parameter, not a R.V (there is no randomness in the parameter as the frequentist point of view claims) and let me call the parameter we are intesrested in as $\theta$. To estimate a parameter you use data that comes from a $R.V$, perform some operation $T$ on the data $\{X\}_i^n$ such that $ T = T(\{X\}_i^n) $. Here the Random Variable is $ T$ not $\theta$. With this clear now the $MSE$ when estimating the parameter $\theta$ is:
$$
MSE = E[(\theta - T)^2] = Var(\theta-T) + E^2[\theta - T] = Var(T) + bias^2(T)
$$
Here I have used the fact that $Var(\theta) = 0$ and $bias(T) = E[\theta - T] = \theta - E[T]$, remember that there is no randomness in $\theta.$ You could come up with more thatn one estimator to estimate $ \theta $, let's say $ T_1 , T_2$. Also let's assume that:
$$
E[T_1] = \theta_1 = \theta => bias(T_1) = = 0 \\
E[T_2] = \theta_2 \neq \theta => bias(T_2) \neq 0
$$
This means that $T_1$ is unbiassed ant $T_2$ is not. So, their $MSE's$ are:
$$
MSE_{T_1} = Var(T_1) + bias^2(T_1) = Var(T_1) \\
MSE_{T_2} = Var(T_2) + bias^2(T_2)
$$
Despite that $T_1$ is unbiassed it could have much more greater variance than $T_2$ so that $ Var(T_1) > Var(T_2) + bias^2(T_2)$ and thus beeing an poorer estimator regarding the $MSE$ performance.
So, to wrap up, if you want to estimate a R.V with a constant, the minimum $MSE$ is reached with the constant equal the expected value of the variable but when you are estimating a parameteer (a constant) with data (samples of a Random Variable) is different, there is no "Expected value" of the parameter, only the parameter value itself. Also, normally you use unbiassed estimators but this doesn't implies that the $MSE$ you obtain is the minimum possible because it could have a greater variance.